Distributed Auctions for Task Assignment and Scheduling in Mobile Crowdsensing Systems

Transcription

1 Dtrbuted Aucton for Tak Agnment and Schedulng n Moble Crowdenng Sytem Zhuojun Duan, We L, Zhpeng Ca Computer Scence, Georga State Unverty, Atlanta, GA, USA Emal: zduan2@tudent.gu.edu, wl28@gu.edu, zca@gu.edu Abtract Wth the emergence of Moble Crowdenng Sytem (MCS), many aucton cheme have been propoed to ncentvze moble uer to partcpate n enng actvte. However, n mot of the extng work, the heterogenety of MCS ha not been fully exploted. To tackle th ue, n th paper, we tudy the jont problem of enng tak agnment and chedulng whle conderng partal fulfllment, attrbute dverty, and prce dverty. We frt elaborately model the problem a a revere aucton and degn a dtrbuted aucton framework. Then, baed on th framework, we propoe two dtrbuted aucton cheme, cotpreferred aucton cheme (CPAS) and tme chedule-preferred aucton cheme (TPAS), whch dffer on the method of tak chedulng, wnner determnaton, and payment computaton. We further rgorouly prove that both CPAS and TPAS can acheve computatonal-effcency, ndvdual-ratonalty, budgetbalance, and truthfulne. Fnally, the mulaton reult valdate the effectvene of both CPAS and TPAS n term of enng tak allocaton effcency, moble uer workng tme utlzaton and utlty, and truthfulne. Index Term Moble crowdenng ytem; truthful aucton; tak agnment; tak chedulng; dtrbuted algorthm. I. INTRODUCTION In the pat few year, the popularty of Moble Crowdenng Sytem (MCS) ha been greatly prompted, n whch enory data can be ubqutouly collected and hared by moble devce n a dtrbuted fahon. Typcally, an MCS cont of a cloud platform, enng tak, and moble uer equpped wth moble devce, n whch the moble uer carry out enng tak and receve monetary reward a compenaton for reource conumpton (e.g., energy, bandwdth, and computaton) and rk of prvacy leakage (e.g., locaton expoure [1]). Compared wth tradtonal mote-cla enor network, MCS can reduce the cot of deployng pecalzed enng nfratructure and enable many applcaton that requre reource and enng modalte beyond the current mote-cla enor procee a today moble devce (martphone (Phone, Sumung Galaxy), tablet (Pad) and vehcle-embedded enng devce (GPS)) ntegrate more computng, communcaton, and torage reource than tradtonal mote-cla enor [2]. The current applcaton of MCS nclude traffc congeton detecton, wrele ndoor localzaton, polluton montorng, etc [2] [4]. There no doubt that one of the mot gnfcant charactertc of MCS the actve nvolvement of moble uer to collect and hare enory data. In other word, for any MCS, one of the mot mportant problem how to get moble uer nvolved n enng tak? Thu, to effectvely ncentvze moble uer to jon moble crowdenng, aucton that a powerful game-theoretcal ncentve mechanm [5] [11] ha been wdely appled to degn market-baed enng tak agnment cheme [3], [12] [18]. Unfortunately, everal crtcal ue are gnored by mot of the extng work: ) n the propoed aucton model [13], [15] [2], a enng tak agned to a moble uer f and only f the tak can be fully completed by the moble uer, whch mpractcal n ome cenaro; n realty, a enng tak mght not be fully completed by one moble uer at a tme (e.g., polluton montorng wthn an area durng a tme perod) a a moble uer avalable workng tme n an MCS lmted; ) heterogenety n MCS not fully explored enng tak may have dfferent requrement n term of locaton, tartng tme, endng tme, type of enor, etc, and moble uer alo vary n ther locaton, avalable tartng tme, avalable endng tme, et of equpped enor, etc; ) due to the aforementoned dverte of tak requrement and uer avalablty, the prce aked by a moble uer to proce dfferent tak are alo dfferent. Motvated by the above obervaton, n th paper, we nvetgate the problem of jonng enng tak agnment and chedulng n MCS wth the followng three conderaton: ) partal fulfllment, whch mean that a enng tak can get agned f t can be partally completed by one or more moble uer n the tme doman; for example, f a tak requet the enory data at a certan locaton from 9:am to 11:am and a moble uer who the only uer n the moble ytem can collect the requred data from 9:3am to 1:3am, the tak wll be agned to the moble uer; ) attrbute dverty, whch ndcate that the mplementaton requrement of tak and the avalablty of moble uer vary n tak attrbute, ncludng locaton, tartng tme, endng tme, and type of enor; and ) prce dverty, whch ay that each moble uer could ak dfferent prce for performng dfferent tak. Notce that the extng aucton cheme [3], [12] [18] do not conder tak chedulng n the tme doman, thu they cannot be appled to olve our problem. Moreover, extendng uch aucton to conder partal fulfllment, attrbute dverty, and prce dverty nontrval. Therefore, degnng a truthful aucton for tak agnment and chedulng whle takng nto account partal fulfllment, attrbute dverty, and prce dverty very challengng. In th paper, to overcome the above challenge, we frt formulate the jont problem of tak agnment and chedulng

2 a a revere aucton, n whch partal fulfllment, attrbute dverty, and prce dverty are condered. Next, we degn a dtrbuted aucton framework, n whch each tak owner ndependently control t local aucton. Baed on uch a framework, we propoe two dtrbuted aucton cheme, cotpreferred aucton cheme (CPAS) that chedule tak accordng to the non-decreang order of moble uer akng prce and tme chedule-preferred aucton cheme (TPAS) that chedule tak accordng to the non-decreang order of moble uer arrval tme. Furthermore, va rgorou theoretcal proof, we how that both CPAS and TAPS can acheve computatonal effcency, ndvdual ratonalty, budget balance, and truthfulne. Fnally, our ntenve mulaton reult confrm the effectvene of the propoed aucton cheme CPAS and TPAS. To um up, our mult-fold contrbuton are a follow: To the bet of our knowledge, we are the frt to etablh a revere aucton model ncorporatng enng tak agnment and chedulng whle conderng partal fulfllment, attrbute dverty, and prce dverty. We degn a dtrbuted aucton framework to allow each tak owner to ndependently proce t local aucton wthout collectng global nformaton n an MCS, reducng communcaton cot. We propoe a cot-preferred aucton cheme (CPAS) to agn each wnnng moble uer one or more ubworkng tme duraton and a tme chedule-preferred aucton cheme (TPAS) to allocate each wnnng moble uer a contnuou workng tme duraton. We perform comprehenve theoretcal analy and prove that both CPAS and TAPS are computatonally effcent, ndvdually ratonal, budget balanced and truthful. The mulaton are well conducted to valdate the performance of CPAS and TPAS n term of allocaton effcency, workng tme utlzaton, utlty, and truthfulne. The ret of th paper organzed a follow. We brefly ummarze the related work n Secton II. Then the ytem model and problem formulaton are preented n Secton III. The cot-preferred aucton cheme and the tme chedulepreferred aucton cheme are propoed n Secton IV and Secton V, repectvely. After evaluatng the performance of the two propoed aucton cheme n Secton VI, we conclude th paper n Secton VII. II. RELATED WORK In th ecton, we manly ummarze the extng aucton mechanm propoed for tak agnment and chedulng n MCS [3], [12] [17], [21]. Note that an aucton can be performed n a centralzed way [12] [17], n whch an auctoneer gather global nformaton and compute the aucton reult. In [12], a uercentrc combnatoral aucton wa degned, n whch each moble uer bd for a et of enng tak wth an akng prce and the crowdourcer am to maxmze t utlty va uer electon. Feng et al. [13] propoed a revere aucton for the platform to mnmze t cot, n whch a enng tak can be done by a martphone f the locaton of the enng tak wthn the ervce coverage of the martphone. Jn et al. [14] degned a ngle-mnded revere combnatoral aucton and a mult-mned revere combnatoral aucton by conderng the qualty of nformaton of moble uer. In [15], va conderng that the enng tak are randomly publhed and the moble uer dynamcally arrve n an MCS, an offlne aucton and an onlne aucton were propoed. Zhang et al. [16] tuded three aucton cheme repectvely correpondng to the followng three cenaro n MCS: ) ngle-requeter ngle-bd model; ) ngle-requeter multple-bd model; and ) multple-requeter multple-bd model. J et al. [17] nvetgated the dcretzaton n crowdenng ytem and degned two aucton-baed ncentve mechanm, n whch each uer ha a unform enng ubtak length. However, n the aucton model of the above work [12] [17], for each enng tak, the requrement of workng tme and type of enor are not taken nto account. To the bet of our knowledge, the only extng work on dtrbuted ncentve mechanm for tak allocaton n MCS [3]. In [3], the author frt formulated the problem of tak electon for moble uer a a non-cooperatve tak electon game and then nvetgated the equlbrum and convergence of the game. In the propoed game, the objectve to maxmze each moble uer utlty by fndng an order to complete one or more enng tak that locate at dfferent place. Dfferent from all the above pror work, n th paper, we propoe to degn dtrbuted truthful aucton cheme for tak agnment and chedulng n MCS whle conderng partal fulfllment, attrbute dverty, and prce dverty. III. SYSTEM MODEL AND PROBLEM FORMULATION A. Sytem Model We conder an MCS contng of a cloud platform, multple enng tak owner (STO), and multple moble uer equpped wth mart devce (MUD). In an MCS, each STO act a a buyer demandng enng tak ervce and each MUD act a a eller offerng enng tak ervce. Both the STO and the MUD can connect to the platform va cloud. The cloud platform allow the STO to perodcally publh ther enng requet. Suppoe that there are m STO and each one own a enng tak to be done. Let Π = {π 1, π 2,, π m } be the et of all STO tak. In th paper, STO tak and tak π are nterchangeable a each STO ha only one tak requet. Each enng tak aocated wth four attrbute: locaton, tartng tme, endng tme, and reource (e.g., camera and gyrocope). Thee four attrbute ndcate the pecfc requrement of enng tak mplementaton and are determned by the STO. Each STO enng tak nformaton, denoted by f π, nclude the four tak attrbute and can be formally repreented a follow: f π = (L π, [α π, β π ], R π ), where L π the requred locaton to perform tak π, α π and β π are repectvely the tartng tme and the endng tme to

3 perform tak π, and R π preent a et of requred reource to mplement tak π. In addton, STO ha budget b to complete tak π per unt tme lot. In other word, STO requet that tak π need to be mplemented at locaton L π durng tme perod [α π, βπ ] by conumng a et of reource R π wth a maxmum unt payment b. On the other hand, there ext n MUD denoted by Γ = {γ 1, γ 2,, γ n }. Each MUD γ j allowed to work for at mot one STO and ha t ntal locaton L γ j, avalable tartng tme α γ j, avalable endng tme βγ j, a et of reource Rγ j embedded nto t mart devce, and an average movng rate λ γ j. Formally, each MUD γ j enng ervce nformaton defned to be f γ j = (Lγ j, [αγ j, βγ j ], Rγ j, λγ j ). Obvouly, to mplement each tak π, a cot of movng from locaton L γ j to locaton Lπ and a cot of conumng reource n R γ j wll be brought to each MUD γ j. Thu, each MUD γ j ha an akng prce vector A j =< a 1j, a 2j,, a mj >, n whch every a j (1 m) an akng prce per unt tme lot ndcatng the cot of movement and reource conumpton to proce tak π. Snce tak π locaton L π may be dfferent from MUD γ j locaton L γ j, MUD γ j hould frt move from L γ j to Lπ and then tart the requred enng tak. Let d(l π, Lγ j ) be the Eucldean dtance between locaton L π and L γ j. Wth movng rate λ γ j, MUD γ j arrve at locaton L π at tme t α j = d(l π,lγ j ) λ γ j + α γ j. For mplcty, we aume that an MUD can tart workng a oon a t arrve at a tak requred locaton. Indeed, th aumpton doe not affect the performance of our propoed model and cheme. Let T j be MUD γ j maxmum avalable workng tme duraton for π and T j be the number of tme lot of tme duraton T j. For each MUD γ j and each tak π, T j can be calculated va the followng x cae: 1) If t α j βπ, tak π fnhed when MUD γ j arrve and thu MUD γ j cannot perform tak π,.e., T j =. 2) If β γ j α π, tak π tart when MUD γ j avalable tme end. Therefore, MUD γ j cannot perform tak π and we have T j =. 3) If α π t α j < βγ j < βπ, t mean that MUD γ j arrve when/after tak π begn and that MUD γ j avalable tme end before tak π endng tme. Accordngly, we have T j = [t α j, βγ j ]. 4) If α π t α j < βπ β γ j, MUD γ j arrve when/after tak π begn and MUD γ j avalable tme end when/after tak π requred workng tme termnate. Thu, the maxmum avalable workng tme duraton T j = [t α j, βπ ]. 5) If t α j < απ < βγ j < βπ, MUD γ j arrve before tak π begn and MUD γ j avalable tme end before tak π endng tme. In th cae, T j = [α π, βγ j ]. 6) If t α j < απ < β π β γ j, MUD γ j arrve before tak π begn and MUD γ j avalable tme end when/after tak π requred workng tme termnate. A a reult, we have T j = [α π, βγ j ]. B. Problem Formulaton When competng for tak π wth other MUD, MUD γ j cheduled an actual workng tme duraton Tj by STO. Correpondngly, denoted by Tj the number of tme lot of Tj. Note that T j the maxmum avalable workng tme duraton of MUD γ j for tak π, thu we obtan the followng relatonhp: ) Tj T j; and ) Tj T j. We ue a -1 bnary varable x j {, 1} to ndcate the tak agnment,.e., γ j procee tak π f and only f x j = 1. Snce each MUD γ j allowed to work for at mot one STO, we have m =1 x j 1. If γ j allocated to perform tak π, γ j can obtan a payment p j from tak owner STO a a reward and receve a utlty U γ j that computed through Eq. (1). U γ j m = m u γ j = x j (p j a j Tj ). (1) =1 =1 In th paper, we conder a practcal cenaro, n whch each STO ndependently control t local tak aucton to determne the wnnng MUD and to chedule ther workng tme. Thu, each STO alo work a an auctoneer of t local tak aucton whch can be formulated to be a revere aucton a preented n Eq. (2). mn x j a j Tj, (2a).t. n x j Tj [α π, β π ], (2b) x j Tj β π α π, (2c) x j {, 1}, 1 j n, T j T j, 1 j n. (2d) (2e) In the above revere aucton Eq. (2), each STO objectve to mnmze the cot for enng tak agnment and chedulng uch that the followng condton can multaneouly hold: ) condton Eq. (2b) requre that the unon of cheduled workng tme duraton cannot exceed the tak tme duraton; ) condton Eq. (2c) ndcate that the total allocated tme lot cannot be more than the number of lot of the tak tme duraton; ) condton Eq. (2d) and (2e) how the range of agnment varable x j and chedule varable T j, repectvely. C. Aucton Economc Properte In an aucton cheme, the followng economc properte are typcally condered [5]: Indvdual-ratonalty. Th tate that no buyer/eller obtan a negatve utlty,.e., n th paper, U γ j for all γ j Γ. Budget-balance. In a double-de aucton, the auctoneer budget the dfference between the total charge collected from all buyer and the total payment pad to all eller. Notce that each STO ha a budget b

4 MUD (Seller) 1. Publh Tak Informaton. 2. Submt Servce Informaton & Prce. 3. Announce Aucton Reult. 4. Reply Fnal Decon. STO (Buyer & Auctoneer) Fg. 1. A dtrbuted aucton framework. Algorthm 1 Cot-Preferred Aucton Scheme for STO 1: Input: f π, [α π, βj π ]. 2: Output: {x j}, {Tj}. 3: Set {x j} = {}, {Tj} = { }, and T u = [α π, βj π ]; 4: repeat 5: Publh enng tak nformaton f π ; 6: Receve enng ervce nformaton {f γ j } and akng prce {a j} from the MUD; 7: Run Alg. 2 to determne potental wnner, chedule workng tme, compute payment, and announce the reult; 8: Collect reple from the MUD, record the value of {x j}, and update T u = T u \ n xjt j; 9: untl T u = or no potental wnner elected. n t ngle-de revere aucton and work a both a buyer and an auctoneer at the ame tme. Therefore, n each STO aucton, budget-balance defned a: x j b Tj n x j p j for all 1 m. Incentve-compatblty. Th alo called truthfulne or trategy-proof, whch ndcate that no bdder can mprove t receved utlty va lyng about t bd prce. In each STO aucton, ncentve-compatblty enure that each MUD γ j Γ can receve a maxmum utlty f and only f t akng prce atfe a j = ā j for all π Π, where ā j denote the true akng prce of MUD γ j for tak π. If an aucton can multaneouly acheve ndvdualratonalty, budget-balance, and truthfulne, t called economc-robut aucton. D. Dtrbuted Aucton Framework In th paper, we propoe to degn dtrbuted aucton cheme contanng four major tage that are preented n Fg. 1. Thee four tage are brefly ummarzed n the followng: Stage 1: Publh Tak Informaton. At the begnnng, each STO publhe t tak nformaton f π and the deadlne of acceptng bd from MUD on the cloud platform. The bd ubmtted by an MUD after the deadlne wll be rejected. Stage 2: Submt Servce Informaton & Prce. After recevng the tak nformaton, each MUD γ j ubmt t ervce nformaton f γ j and akng prce a j to STO f t ntereted n tak π. Stage 3: Announce Aucton Reult. Each STO collect ervce nformaton and akng prce from the MUD, chedule workng tme, decde the potental wnner, and payment. Then, each STO announce the aucton reult and a deadlne of ubmttng fnal decon to the MUD who have ubmtted nformaton and prce. Each MUD hould reply t fnal decon to the STO who have choen t a a potental wnner before the deadlne. Stage 4: Reply Fnal Decon. If MUD γ j choen a a potental wnner by one or more STO, t hould reply t fnal decon to the STO who have choen t a a potental wnner. From the above four tage, one can ee that an STO may be rejected by the MUD. Thu, to complete the enng tak, each STO contnue to conduct t revere aucton to chedule the remanng unagned workng tme lot n a mult-round manor untl t tak tme duraton ha been fully cheduled or no potental wnnng MUD can be elected. Meanwhle, f an MUD uccefully cheduled to a tak, t ext the aucton; otherwe, t contnue to compete for workng untl no tak nformaton publhed. Furthermore, under the propoed aucton framework, two dfferent polce can be ued to perform tak agnment and chedulng: ) cot-preferred polcy: the STO determne the potental wnner accordng to the non-decreang order of the MUD akng prce; and ) tme chedule-preferred polcy: the STO determne the potental wnner baed on a frt-come-frt-erve manor. The adopton of the polcy determned through MUD negotaton before conductng the aucton. The aucton cheme correpondng to the two polce are detaled n Secton IV and V, repectvely. IV. COST-PREFERRED AUCTION SCHEME In th ecton, a Cot-Preferred Aucton Scheme termed CPAS propoed, n whch each STO greedly perform enng tak agnment and chedulng accordng to the nondecreang order of the MUD akng prce. The tage of CPAS for each STO outlned n Algorthm 1. Snce the aucton cheme CPAS performed n a multround manor and the aucton proce of each round the ame, we jut demontrate the aucton proce of a round n the followng part of th ecton. A. Informaton Publcaton & Collecton At the begnnng of an aucton, each STO publhe t tak nformaton f π on the cloud platform. After obtanng all the STO tak nformaton, each MUD γ j ubmt t ervce nformaton f γ j and akng prce a j to STO. Note that an MUD could be ntereted n more than one enng tak and end t ervce nformaton and akng prce to the correpondng STO at the ame tme.

5 B. Potental Wnner Determnaton and Payment Calculaton When STO receve ervce nformaton {f γ j } and akng prce {a j } from one or more MUD, baed on {f γ j }, {a j}, f π, and b, STO form a et of avalable MUD a Γ c (π ) = {γ j (T j T u ), R γ j Rπ, and a j b }, n whch T u denote the un-cheduled tme duraton for tak π at the current round of aucton. Th computaton mplemented n lne 2-7 of Algorthm 2. 1) Potental Wnner Determnaton: Intally, the et of potental wnner W (π ) =. To chedule workng tme, STO frt ort all the avalable MUD n Γ c (π ) n a nondecreang order n term of ther akng prce and get a orted et Γ c (π ) (ee lne 8 n Algorthm 2). Next, STO can the avalable MUD n Γ c (π ) and allocate uncheduled tme lot n a greedy manor. Specfcally peakng, f MUD γ j current avalable workng tme duraton (T j T u) ha not been fully cheduled to other avalable MUD,.e., (T j T u) ( Tj ) (T j T u), MUD γ can γ j W (π ) be choen a a potental wnner and agned a et of tme lot that ha not been allocated to current potental wnner n W (π ),.e., Tj = (T j T u)\(t j T u ( Tj )) γ j W (π ) (ee lne 9-14 n Algorthm 2). 2) Payment Calculaton: After completng tak chedulng, STO compute the payment for each potental wnnng MUD γ j va dentfyng γ j crtcal neghbor, whch defned to be the MUD γ k n Γ c (π ) where f a j hgher than a k, γ j can not be cheduled. Dfferent from the prevou work [13], [15] [2] n whch each wnner ha only one crtcal neghbor, each wnnng MUD γ j n the aucton CPAS ha one or more crtcal neghbor becaue the tme lot of Tj could be cheduled to one or more other avalable MUD f MUD γ j doe not jon the aucton (ee lne of Algorthm 2). Thu, the payment calculated accordng to every crtcal neghbor of wnner γ j. In order to fnd the crtcal neghbor, STO ort all the MUD n Γ c γ j (π ) = Γ c (π )\γ j n the non-decreang order n term of ther akng prce, elect wnner agan n the orted et Γ c γ j (π ), and chedule workng tme to them. Any MUD γ k a crtcal neghbor of MUD γ j f ther allocated tme duraton are overlappng,.e., Tk T j, where Tk the tme duraton agned to MUD γ k and Tj record the remanng tme duraton n T j that not allocated to other. So the correpondng crtcal payment a k Tk Tj. But, f no crtcal neghbor found for MUD γ j, t crtcal payment STO budget b Tj. Then, STO announce the aucton reult {Tj } and {p j} to the MUD. Remark: Va Algorthm 2, each potental wnnng MUD can receve a workng tme duraton Tj that contan one or more ub-tme duraton. For example, the tme duraton of tak π from 1:pm to 5:pm, wnner γ j workng tme duraton Tj = {[2:pm, 3:pm], [4:3pm, 5:pm]} contanng two ub-tme duraton, and the number of workng tme lot Tj = 9 mnute. Algorthm 2 Cot-Preferred Tak Schedulng & Prcng for Tak π Input: f π, b, T u, Γ, {f γ j }, {aj}. Output: W (π ), {Tj}, {p j}. 1: Set Γ c (π ) =, W (π ) =, {Tj} = { }, and {p j} = {}; 2: for each γ j Γ wth ubmtted f γ j and aj do 3: Calculate T j; 4: f (T j T u ), R γ j Rπ, and a j b then 5: Γ c (π ) = Γ c (π ) γ j; 6: end f 7: end for 8: Sort all MUD n Γ c (π ) n non-decreang order baed on {a j} and obtan the orted et Γ c (π ); 9: for j = 1 to Γ c (π ) do 1: f (T j T u ) ( T j ) (Tj T u ) then γ j W (π ) 11: W (π ) = W (π ) γ j; 12: Tj = (T j T u )\(T j T u ( γ j W (π ) T j )); 13: end f 14: end for 15: for each γ j W π do 16: Set {Tk} = { } and Tj = Tj; 17: Sort all the MUD n Γ c (π )\γ j n a non-decreang order baed on {a k } and obtan the orted et Γ c γ j (π ); 18: Set k = 1 and W γj (π ) = ; 19: whle k Γ c γ j (π ) and Tj do 2: f (T k T u ) ( T j ) (T k T u ) then γ j W γj (π ) 21: W γj (π ) = W γj (π ) γ k, 22: Tk = (T k T u )\(T k T u ( 23: f T k T j then 24: p j = p j + a k T k T 25: T j = T j \(T k T 26: end f 27: end f 28: k = k + 1; 29: end whle 3: f Tj then 31: p j = p j + b T 32: end f 33: end for j. C. Fnal Servce Decon j, j ); γ j W γj (π ) T j )); Each MUD γ j ndependently make t ervce decon when t obtan the aucton reult from the STO. Let Π(γ j ) be the et of tak of whch ther owner elect MUD γ j a a potental wnner, whch defned a Π(γ j ) = {π γ j W (π ) and π Π}. The decon proce decrbed a follow. If Π(γ j ) =, MUD γ j a loer n each STO local aucton cheme CPAS, doe not need to end a reply, and reman n the aucton untl no tak requet publhed. If Π(γ j ) = 1, MUD γ j a potental wnner n an STO locaton aucton, accept the ervce requet, and ext the aucton. If Π(γ j ) > 1, MUD γ j elected a a potental wnner by multple STO and accept the STO who brng γ j

6 the maxmum utlty. That, the accepted tak requet π decded a π = arg max {(p hj a hj Thj )}. π h Π(γ j) Then, MUD γ j ext the aucton. Fnally, each STO et the value of {x j } baed on W (π ) and the MUD reple. D. Property Analy In th ubecton, we theoretcally analyze the performance of aucton mechanm CPAS n term of computatonal effcency, ndvdual-ratonalty, budget-balance, and truthfulne. Lemma 1: The cot-preferred chedulng cheme, Algorthm 2, can termnate wthn O(n 2 log(n)). Proof: From lne 2 to lne 7, the runnng tme of formng et Γ c (π ) at mot n that the number of MUD n et Γ. In lne 8, ortng the MUD n Γ c (π ) cot at mot n log(n) tme. The potental wnner determnaton, n lne 9-14, ha a tme complexty of O(n). Smlarly, we know that the ortng proce of lne 17 ha a tme complexty of O(n log(n)) and that fndng crtcal neghbor termnate wthn O(n). Addtonally, the for loop from lne 15 to lne 33 contan at mot n teraton and can end wthn O(n 2 log(n)). To um up, the tme complexty of Algorthm 2 O(n 2 log(n)). Theorem 1: The propoed aucton cheme CPAS computatonally effcent wth a tme complexty of O(n 3 log(n)). Proof: From Algorthm 1, one can ee that each STO top f and only f ether of the two condton atfe: ) T u = ; and ) no potental wnner elected. Let u conder the wort cae for any STO : STO pck only one potental wnner at each round but rejected by the potental wnner. Under th tuaton, the potental wnner defntely accept another STO tak requet and then ext the aucton. Thu, after at mot n round, STO end t aucton a no potental wnner can be choen. From Lemma 1, we obtan the concluon that the tme complexty of CPAS O(n 3 log(n)). Theorem 2: The aucton cheme CPAS enure ndvdualratonalty for all MUD. Proof: If MUD γ j a loer for all the tak n an aucton, we have m x j = and U γ j =. =1 If MUD γ j a wnner for tak π n an aucton, x j = 1 and Tj >. Due to the defnton of crtcal neghbor and et Γ c (π ), we have a k a j for γ j every crtcal neghbor γ k and b a j for STO, ndcatng that p j a j Tj (ee lne 24 and lne 31 of Algorthm 1). Therefore, U γ j = m u γ j = =1 m x j (p j a j Tj ). =1 Theorem 3: The aucton cheme CPAS budget-balanced for all STO Proof: From Algorthm 1, t can be een that all the potental wnner are elected from et Γ c (π ) and a j b for all γ j Γ c (π ). In addton, from lne 24 and lne 31 of Algorthm 1, we have b p j for each wnner γ j. Thu, x j b Tj n x j p j,.e., CPAS acheve budgetbalance for each STO. Lemma 2: In each STO aucton CPAS, f MUD γ j elected a a potental wnner wth a prce a j, t can tll be a potental wnner wth a maller prce a 1 j < a j and Tj T j 1 1, where Tj the agned workng tme duraton correpondng to a 1 j. Proof: Suppoe that po(a 1 j ) and po(a j) are the poton of MUD γ j n the orted et Γ c (π ) when bddng wth a 1 j and a j, repectvely. Snce a 1 j < a j, po(a 1 j ) po(a j ). From the method of chedulng and prcng (ee Algorthm 2), t een that MUD γ j ubmttng a 1 j can be uccefully cheduled a tme duraton Tj 1 and T j T j 1. Theorem 4: The aucton cheme CPAS guarantee truthfulne for all MUD. Proof: To prove th theorem, t equvalent to prove that n each STO local aucton CPAS, each MUD γ j Γ cannot enhance t utlty by ubmttng an akng prce a j ā j. Th can be analyzed through the followng cae. Cae 1: a j < ā j (or a j > ā j ) and MUD γ j loe the aucton wth both a j and ā j. In th cae, γ j utlty receved from STO aucton zero. Cae 2: a j < ā j and MUD γ j can wn the aucton wth both a j and ā j. Accordng to Lemme 2, we have T j T j and T j T j, where T j and T j repectvely denote the agned tme duraton and the number of tme lot of T j correpondng to ā j. Accordngly, the payment p j can be re-computed va two part: ) the payment p j pad for tme duraton T j that the ame for both ā j and a j ; and ) payment p j pad for tme duraton Tj \ T j, n whch a j Tj \ T j p j ā j Tj \ T j a a j a k ā j for γ j every crtcal neghbor γ k. Correpondngly, the receved utlty u γ j = p j ā j Tj = ( p j ā j T j ) + ( p j ā j Tj \ T j ). Snce a j Tj \ T j p j ā j Tj \ T j, we have ( p j ā j Tj \ T j ). A a reult, we obtan u γ j = p j ā j Tj p j ā j T j,.e., MUD γ j cannot get a hgher utlty by bddng a j. Cae 3: a j < ā j and MUD γ j wn wth a j but loe wth ā j. In th cae, we know that ā j hgher than t crtcal neghbor akng prce {a k } or hgher than STO budget b. That, ā j T j p j. Therefore, we have u γ j = p j ā j T j. Cae 4: a j > ā j and MUD γ j wn wth ā j but loe wth a j. In th cae, u γ j = whch cannot be hgher than the utlty correpondng to ā j. Cae 5: a j > ā j and MUD γ j wn the aucton wth both ā j and a j. Smlar to the analy of Cae 2, we have Tj T j and T j T j. In addton, payment p j cont of two part: ) the payment p j pad for tme duraton Tj that the ame for both ā j and a j ; and ) the payment p j pad for tme duraton T j \ T j, n whch ā j T j \ T j p j a j T j \T j a ā j a k a j for γ j every crtcal neghbor γ k. Thu, the receved utlty u γ j = p j ā j Tj (p j ā j Tj ) + ( p j ā j T j \ T j ); that, MUD γ j utlty cannot be enhanced by ubmttng a j > ā j.

7 Algorthm 3 Tme Schedule-Preferred Aucton Scheme for STO 1: Input: f π, [α π, βj π ]. 2: Output: {x j}, {Tj}. 3: Set {x j} = {}, {Tj} = { }, and T u = [α π, βj π ]; 4: repeat 5: Publh enng tak nformaton f π ; 6: Receve enng ervce nformaton {f γ j } and akng prce {a j} from the MUD; 7: Run Alg. 4 to determne potental wnner, chedule workng tme, compute payment, and announce the reult; 8: Collect reple from the MUD, record the value of {x j}, and update T u = T u \ n xjt j; 9: untl T u = or no potental wnner elected. In ummary, each STO aucton CPAS truthful for all MUD. Furthermore, from Subecton IV-C, we can conclude that each MUD γ j cannot ncreae the value of max {(p j π Π(γ j) a j Tj )} va cheatng on t akng prce a j for each tak π. Therefore, the aucton cheme CPAS can acheve truthfulne for all MUD. V. TIME SCHEDULE-PREFERRED AUCTION SCHEME Notce that n the aucton CPAS, a wnnng MUD workng tme duraton contan one or more ub-tme duraton. To allocate one ngle contnuou tme duraton to each MUD, we propoe a tme chedule-preferred aucton cheme termed TPAS, n whch an STO frt chedule the MUD baed on a frt-come-frt-erve manor n the tme doman and then compute the payment for each wnnng MUD. The tage of TPAS for each STO are outlned n Algorthm 3. A. Informaton Publcaton & Collecton In th tage, each STO publhe t tak nformaton f π on the platform. Then, each MUD ubmt t ervce nformaton f γ j and akng prce a j to STO f the MUD ntereted n tak π. B. Potental Wnner Determnaton & Payment Calculaton After obtanng the ubmtted ervce nformaton, each STO compute the et of avalable MUD, Γ t (π ), a follow: Γ t (π ) = {γ j (T j T u ) and R γ j Rπ }. 1) Potental Wnner Determnaton: Accordng to the frtcome-frt-erve polcy, each STO greedly agn a workng tme duraton to each avalable MUD γ j accordng to the non-decreang order n term of the MUD arrval tme t α j = { d(lπ, Lγ j ) + α γ λ γ j } untl no avalable MUD can be j elected or t unagned workng tme duraton T u become empty. More pecfcally, n order to chedule an a long contnuou workng tme duraton a poble, STO agn each avalable MUD γ j Γ t (π ) a tme duraton from γ j pror MUD endng workng tme to the tme mn{β π, βγ j } f th tme duraton unagned. The peudo-code of the chedulng cheme preented n Algorthm 4. Algorthm 4 Tme Schedule-Preferred Tak Schedulng for Tak π 1: Input: f π, T u, Γ, {f γ j }. 2: Output: {Tj}. 3: Set Γ t (π ) = and {Tj} = { } for γ j Γ(π ); 4: for each γ j Γ wth ubmtted f γ j and aj do 5: Calculate t α j and T j; 6: f (T j T u ) and R γ j Rπ then 7: Γ t (π ) = Γ t (π ) γ j; 8: end f 9: end for 1: Sort all MUD n Γ t (π ) n the non-decreang order baed on {t α j} and get the orted et Γ t (π ); 11: Set Start = α π ; 12: for j = 1 to Γ t (π ) do 13: f Start < mn{β γ j, βπ } and (T j Tj) u ( j 1 T j ) (T j T u j) then 14: T j = [Start, mn{β γ j, βπ }]; 15: Start = mn{β γ j, βπ }. 16: end f 17: end for j =1 2) Payment Calculaton: After completng tak chedulng, each STO frt ort all the avalable MUD akng prce n the non-decreang order. Wthout lo of generalty, n et Γ t (π ), we mply aume that a 1 a 2 a Γ t (π ). Then, each STO earche for a maxmum ndex k π uch that a k π b < a k π +1, ntalze the et of potental wnner W (π ) =, and determne wnner accordng to the followng two cae. Cae 1: Tk. If γ π j Γ t (π ), 1 j k π, and Tj, et W (π ) = W (π ) γ j and p j = b Tj. Moreover, n th cae, b the crtcal prce of all he MUD n STO aucton. Cae 2: Tk =. If γ π j Γ t (π ), 1 j < k π, and Tj, et W (π ) = W (π ) γ j and p j = a k π Tj. In th cae, MUD γ k π and a k π are the crtcal neghbor and the crtcal prce of all the MUD n STO aucton, repectvely. Next, each STO notfe the MUD of the aucton reult. C. Fnal Servce Decon When learnng the aucton reult, each MUD γ j make t fnal decon ung the method the ame a that of Subecton IV-C. Fnally, the reult of {x j } can be obtaned. D. Property Analy In th ubecton, we theoretcally prove the performance of the aucton cheme TPAS n term of computatonal effcency, ndvdual-ratonalty, budget-balance, and truthfulne. Lemma 3: The computatonal complexty of the chedulng cheme Algorthm 4 O(n log(n)). Proof: From lne 4 to lne 9, the contructon of et Γ t (π ) can be done wthn O(n). In lne 1, the ortng proce can be completed wthn O(n log(n)). From lne 12 to lne

8 17, the chedulng proce contan at mot n teraton and each teraton ha a tme complexty of O(1). Therefore, the computatonal complexty of Algorthm 4 O(n log(n)). Lemma 4: The computatonal complexty of payment calculaton n the aucton cheme TPAS O(n). Proof: To compute the payment, each STO ha to earch for a maxmum ndex k π by cannng et Γ t (π ). A Γ t (π ) n, the computaton complexty of payment calculaton O(n). Theorem 5: The propoed aucton cheme TPAS acheve computatonal effcency wth a tme complexty of O(n 2 log(n)). Proof: From Lemma 3 and 4, and the analy of Theorem 1, th theorem can be proved. Theorem 6: The propoed aucton cheme TPAS ndvdually-ratonal for all MUD. Proof: When all STO aucton end, there are two cae for each MUD γ j : If m =1 x j =, we have p j = and Tj = for each π Π. Thu, U γ j =. If π Π, x j = 1, we have U γ j = u γ j = p j a j Tj a p j a j Tj and T j >. Therefore, we can conclude that TPAS acheve ndvdual ratonalty for all MUD. Theorem 7: The propoed aucton cheme TPAS budgetbalanced for all STO. Proof: If enng tak π uccefully agned to one or more MUD, we have n x j 1, x j Tj >, and p j b Tj for every wnnng MUD γ j. Thu, for STO, we have n x j b Tj n x j p j, ndcatng that TPAS can enure budget-balance for all STO. Lemma 5: For each STO, the chedulng reult {Tj } of Algorthm 4 are ndependent of all MUD akng prce {a j }. Proof: From lne 12 to lne 16 of Algorthm 4, one can ee that the computaton of γ j workng tme duraton Tj doe not depend on t akng prce a j. Therefore, th theorem hold. Lemma 6: In each STO locaton aucton TPAS, f MUD γ j a potental wnner wth bddng a prce a j, t can alo become a potental wnner wth a maller prce a 1 j < a j. Proof: When MUD γ j ubmt a maller prce a 1 j, γ j poton n et Γ t (π ) change from j to j 1. Snce a 1 j < a j, we have j 1 j k π. In addton, accordng to Lemma 5, the agned workng tme duraton Tj reman the ame for MUD γ j. A a reult, γ j can be tll elected a a potental wnner by STO for tak π. Theorem 8: The propoed aucton cheme TPAS can acheve truthfulne for all MUD. Proof: Provng th theorem equvalent to prove that n each STO local aucton TPAS, each MUD γ j Γ cannot mprove t utlty u γ j by akng for a prce a j ā j, n whch there are fve cae to be condered for each MUD γ j. Cae 1: a j < ā j (or a j > ā j ) and MUD γ j loe the aucton wth both a j and ā j. In th cae, γ j utlty receved from STO zero. Cae 2: a j < ā j and MUD γ j wn the aucton wth both a j and ā j. Through Lemma 5, we know that the agned tme duraton Tj for MUD γ j wth both a j and ā j. In addton, from the property of the prcng method n TPAS and Lemma 6, we have a j < ā j a k π b,.e., a j Tj < ā j Tj p j. Therefore, the utlty alo reman the ame,.e., u γ j = p j ā j Tj. Cae 3: a j < ā j and MUD γ j wn wth a j but loe wth ā j. In th cae, ā j hgher than the crtcal prce a k π or hgher than STO budget b. Thu, we have ā j Tj p j accordng to the prcng method n Subecton V-B2. A a reult, the utlty u γ j = p j ā j Tj. Cae 4: a j > ā j and MUD γ j wn wth ā j but loe wth a j. In th cae, u γ j = whch cannot be hgher than the utlty correpondng to ā j. Cae 5: a j > ā j and MUD γ j wn the aucton wth both ā j and a j. Smlar to Cae 2, we have the followng relatonhp: ) Tj for MUD γ j wth both a j and ā j from Lemma 5; and ) ā j < a j a k π b due to the property of the prcng method n TPAS and Lemma 6. Thu, the utlty unchanged,.e., u γ j = p j ā j Tj. The above fve cae ndcate that each STO aucton truthful for all MUD. Moreover, from Subecton IV-C, one can ee that each MUD γ j cannot ncreae the value of max {(p j a j Tj )} va cheatng on t akng prce a j π Π(γ j) for each tak π. Therefore, the aucton cheme TPAS can enure truthfulne for all MUD. Remark: In the aucton cheme TPAS, the proce of tak chedulng ndependent of the MUD akng prce. A a reult, an MUD that ha been agned a non-empty tme duraton cannot wn the aucton f the MUD akng prce hgher than the correpondng crtcal prce. In fact, any prcendependent chedulng algorthm can be appled n TPAS to obtan {Tj }, wthout any mpact on truthfulne for the MUD. A. Smulaton Settng VI. PERFORMANCE EVALUATION We evaluate the performance of the cot-preferred aucton cheme (CPAS) and tme chedule-preferred aucton cheme (TPAS) baed on a ynthetc data et. The number of enng tak vare from 5 to 3, and the number of MUS vare from 5 to 15. The locaton of all enng tak are randomly and unformly dtrbuted wthn a rectangular area of 6km 6km. The locaton of all MUD are et accordng to two knd of dtrbuton: ) unform: the locaton are unformly deployed wthn the rectangular area of 6km 6km at random; and ) hotpot: the locaton of each tak vewed a a crclecentered hotpot wth a radu of.7km, and the MUD randomly locate wthn thee hotpot area. We conder 1 type of enor and the number of each type of enor one. Each enng tak requet a number of dfferent enor,

9 Ave. tak' allocaton effcency (m=1) Ave. tak' allocaton effcency (m=2) Ave. MUD' workng tme utlzaton Ave. MUD' utlty CPAS-Unform TPAS-Unform CPAS-Hotpot TPAS-Hotpot CPAS-Unform TPAS-Unform CPAS-Hotpot TPAS-Hotpot 1.8 CPAS-Unform TPAS-Unform CPAS-Hotpot TPAS-Hotpot 3 25 CPAS-Unform TPAS-Unform CPAS-Hotpot TPAS-Hotpot Number of MUD Number of MUD Number of tak Number of tak Fg. 2. Average tak allocaton effcency (m=1). Fg. 3. Average tak allocaton effcency (m=2). Fg. 4. Average MUD workng tme utlzaton. Fg. 5. Average MUD utlty. whch a random number unformly pcked from [3, 1]; mlarly, each MUD equpped wth a number of dfferent enor, whch a random number unformly choen from [1, 1]. In the mulaton, the unt tme lot one mnute and the longet tme duraton 5 hour. More pecfcally, each enng tak (and each MUD avalable tme) randomly begn at or after 1:pm and end at or before 5:pm; that, for any tak (and any MUD), the workng tme duraton at mot 5 hour contanng 3 tme lot. Each STO budget (and each MUD akng prce) an nteger that unformly elected from [1, 25] at random. Suppoe that all the MUD are pedetran wth moble devce, o the movng rate of each MUD randomly and unformly choen wthn [4.5, 5.4]km/hour [22]. We ue the followng metrc to evaluate the performance of the two propoed aucton cheme. Allocaton Effcency. The allocaton effcency of a enng tak defned to be the rato of the total number of agned workng tme lot to the number of requeted workng tme lot. Formally, the average allocaton effcency of all enng tak calculated a 1 m x m ( j Tj ). =1 β π απ Workng Tme Utlzaton. The workng tme utlzaton of an MUD the rato of the number of agned workng tme lot to the number of avalable workng tme lot. Thu, the average workng tme utlzaton of all MUD computed by m 1 n n ( x j Tj =1 β γ j ). αγ j Utlty. We alo compare the average utlty of all MUD. Truthfulne. At each tme, we randomly pck an MUD, et fake akng prce to t, and examne t receved utlty when bdng truthfully and untruthfully. B. Smulaton Reult and Analy We frt check the average allocaton effcency of all enng tak wth the number of STO changng from 1 to 2 and the number of MUD ncreang from 5 to 15 under the unform and hotpot dtrbuton. The reult are preented n Fg. 2 and Fg. 3. A hown n Fg. 2 and Fg. 3, the average allocaton effcency ncreae when the number of MUD ncreae. Th becaue f more MUD partcpate n the aucton, each STO can fnd more MUD to mplement t enng tak. Bede, n both Fg. 2 and Fg. 3, CPAS perform better than TPAS. The reaon le n the followng two apect: ) CPAS agn each MUD one or more workng tme duraton whle TPAS agn each MUD at mot one workng tme duraton; ) the workng tme chedulng of TPAS ndependent of the MUD akng prce, leadng to that ome MUD who have been cheduled a workng tme duraton may be loer f ther prce are hgher than the crtcal prce. In other word, n CPAS, a larger porton of requred workng tme duraton can be cheduled to the MUD, gettng a hgher allocaton effcency for each STO. Moreover, CPAS under the unform dtrbuton can acheve a hgher allocaton effcency than CPAS under the hotpot dtrbuton. The ame tuaton alo occur to TPAS. Th due to the fact that under the hotpot dtrbuton, the number of MUD that are near to any STO reduced, ndcatng that the number of avalable MUD become maller for each STO. Next, we analyze the average workng tme utlzaton and the average utlty of all MUD wth 11 MUD and the number of enng tak ncreang from 5 to 3. The reult of the average workng tme utlzaton and the average utlty are plotted n Fg. 4 and Fg. 5, repectvely. From Fg. 4 and Fg. 5, we obtan the followng two obervaton. Frt, one can ee that the average workng tme utlzaton and the average utlty gradually ncreae a more and more enng tak are publhed, becaue t become le compettve for the MUD to be agned more workng tme lot when more tak are avalable. Second, CPAS (repectvely TPAS) under the unform dtrbuton obtan a larger average workng tme utlzaton and a hgher average utlty than CPAS (repectvely TPAS) under the hotpot utlzaton. The reaon that an MUD uually elected by an STO who near to t but under the hotpot dtrbuton, and the number of STO who are near to any MUD decreaed,.e., the probablty of becomng a wnner reduced for each MUD. Furthermore, we verfy truthfulne of CPAS and TPAS, n whch there are 2 tak and 11 MUD under each dtrbuton cenaro. In the mulaton, we randomly pck one MUD at a tme, et fake akng prce to the pcked MUD,

10 Utlty (Unform) Utlty (Unform) MUD 3 MUD 25 MUD 36 MUD 89 MUD 99 Utlty (Hotpot) Utlty (Hoppot) Utlty wth truthful akng prce Utlty wth untruthful akng prce Utlty wth truthful akng prce Utlty wth untruthful akng prce Natonal Natural Scence Foundaton of Chna under Grant No ID of MUD Fg. 6. MUD truthfulne n CPAS under unform dtrbuton Utlty wth truthful akng prce Utlty wth untruthful akng prce MUD 2 MUD 27 MUD 29 MUD 56 MUD 58 ID of MUD Fg. 8. MUD truthfulne n TPAS under unform dtrbuton MUD 3 MUD 25 MUD 36 MUD 89 MUD 99 ID of MUD Fg. 7. MUD truthfulne n CPAS under hotpot dtrbuton Utlty wth truthful akng prce Utlty wth untruthful akng prce MUD 2 MUD 27 MUD 29 MUD 56 MUD 58 ID of MUD Fg. 9. MUD truthfulne n TPAS under hotpot dtrbuton. and compare t receved utlte when truthful bddng and untruthful bddng. We totally elect fve dfferent MUD under each dtrbuton cenaro and preent the reult n Fg.6-9. Notce that all the elected MUD cannot receve hgher utlte va cheatng on akng prce. For example, n Fg.6, the elected MUD are the 3rd MUD, the 25th MUD, the 36th MUD, the 89th MUD, and the 99th MUD. More pecfcally, the tuaton of the fve MUD are llutrated n the followng: ) the 3rd MUD a wnner when truthful bddng but a loer when cheatng; ) the 25th MUD wn the aucton when bddng truthfully and untruthfully, but t utlty reduced when bddng untruthfully; ) the 36th MUD receve the ame utlty when bddng truthfully and untruthfully; v) the 89th MUD obtan a zero utlty wth truthful prce but a negatve utlty wth fake prce; and v) the 99th MUD utlty reduced to a negatve value when cheatng. VII. CONCLUSION To motvate moble uer to jon enng tak n MCS, we propoe a revere aucton model and two novel dtrbuted aucton cheme, CPAS and TPAS, for tak agnment and chedulng. 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